and S. Boyd

Simultaneous Routing and Resource Allocation via Dual Decomposition

L. Xiao

, M. Johansson

and S. Boyd

∗

Information Systems Laboratory, Department of Electrical Engineering Stanford University, Stanford, CA 94305

e-mail: {lxiao,boyd}@stanford.edu

†

Department of Electrical Engineering and Computer Science University of California, Berkeley, CA 94720

e-mail: [email protected]

Abstract

In wireless networks, the optimal routing and resource allocation problems are coupled together through link capacities, which influence data routing and are deter- mined by resource (e.g., power and bandwidth) alloca- tion. We formulate the problem of simultaneous rout- ing and resource allocation for wireless data networks as a convex optimization problem and exploit the sep- arable structure of the problem via dual decomposition to develop efficient solution methods.

1 Introduction

Many relevant routing problems for data networks can be formulated as convex multicommodity network flow problems (e.g., [1]), for which many efficient solution methods have been developed (e.g., [2, 3]). The op- timal routing tables are highly dependent on the ca- pacities of the communication links, which usually are assumed fixed in the routing literature. The emergence of wireless networks gives new perspectives on optimal routing problems, in particular, when they are coupled with resource allocation problems (e.g., [4, 5]). The capacities of wireless channels are determined by the channel characteristics and communications resources, such as transmit powers, bandwidths, or time-slot frac- tions allocated to the channels. Adjusting the resource allocation within the network can change the capaci- ties of individual links, influence the optimal routing of data flows and alter the total utility of the network.

In this paper, we consider the simultaneous routing and resource allocation (SRRA) problem for wireless networks. We use a capacitated multicommodity flow model to describe the data flows in the network. Ig- noring the detailed transmission protocols and mecha- nisms, the network flow variables should be interpreted as average flows in bits per second. We assume that the capacity of a wireless link is a concave and increasing function of the communications resources allocated to

the link, and the communications resources for groups of links are limited. These assumptions allow us to formulate the SRRA problem as a convex optimization problem over the network flow variables and the com- munications variables. We exploit the special structure of the SRRA problem via dual decomposition and de- rive efficient algorithms for solving the dual problem.

2 Network flow model

We use the standard directed-graph model for network topology and a multicommodity flow model for the av- erage behavior of data transmission.

2.1 Network topology

We model the topology of a communication network by a directed graph. In this model, a collection of nodes, labeled by n = 1, . . . , N , can send, receive and relay data through communication links. A communication link is an ordered pair (i, j) of distinct nodes. The presence of a link (i, j) means that the network can support data flow from the start node i to the end node j. We label all the links with integers l = 1, . . . , L.

We define O(n) as the set of links that are outgoing from node n, and I(n) as the set of links that are incoming to node n.

The network topology can be represented by its node- link incidence matrix. Assume that the network has N nodes and L links, then the entries a

of the incidence matrix A ∈ R

is associated with node n and link l via

a

=

 



1, if l ∈ O(n)

−1, if l ∈ I(n) 0, otherwise.

Hence, each column of A describes a link, and it has

exactly two nonzero entries: one equal to 1 and the

other equal to −1, indicating the start and end nodes

of the link. Each row of A describes all links incident

to a node: the +1 entries indicate outgoing links, and

the −1 entries indicate incoming links.

2.2 Multicommodity network flows

In the multicommodity flow model, each node can send (different) data to many destinations and receive data from many sources (but we don’t consider multicast).

We identify the flows in the network by their destina- tions, i.e., flows with the same destination are consid- ered as one single commodity regardless of their ori- gins. We label the destination nodes as d = 1, . . . , D, where D ≤ N (we can always label the destinations as the first D nodes). For each destination d, we define a source vector s

∈ R

, whose entry s

(n 6= d) denotes the nonnegative flow injected into the network at node n and destined for node d. By the flow conser- vation law, we define the sink flow at the destination

s

= − X

n,n6=d

s

, (1)

where the summation is over all n except for n = d.

At each intermediate node, flows with the same desti- nation are aggregated and transmitted according to the routing table, which possibly splits the flow among out- going links. On each link l, we let x

be the amount of flow destined for node d and call x

∈ R

the flow vector with destination d. At each node n, components of the flow vector and the source vector with the same destination satisfy the following flow conservation law:

X

l∈O(n)

x

− X

l∈I(n)

x

= s

, d = 1, . . . , D.

They can be compactly written as

Ax

= s

, d = 1, . . . , D, (2) where A is the incidence matrix. Note that (2) im- plies (1) since 1

A = 0, where 1 is the vector of ones.

Finally, we impose the link capacity constraints. Let c

be the capacity of link l and t

= P

d

x

be the total amount of traffic on link l. We must have t

≤ c

. In summary, our network flow model imposes the fol- lowing group of constraints on the network flow vari- ables x

, s

and t:

Ax

= s

, d = 1, . . . , D x

º 0, d = 1, . . . , D t

= P

d

x

, l = 1, . . . , L t

≤ c

, l = 1, . . . , L

(3)

where º means component-wise inequalities. We will use x to denote the collection of flow vectors x

and use s to denote the collection of source vectors s

. In convex multicommodity network flow problems, the capacities c

are fixed and one is to minimize a convex cost function of the network variables subject to the constraints (3); see, e.g., [2, 3]. In a wireless network, however, the link capacities c

are typically functions of the communications resources allocated to the links.

These capacity constraints will be described next.

3 Communications model

Now we consider the wireless communication system that supports the data network. In such a system, the capacities of the individual links (channels) depend on the media access scheme and the selection of cer- tain critical parameters, such as transmit powers and bandwidths or time-slot fractions allocated to individ- ual or groups of channels. We refer to these critical communications parameters collectively as communi- cations variables, and denote the vector of communi- cations variables by r. We assume that the medium access methods and coding and modulation schemes of the communication system are fixed, but that we can optimize over the communications variables r.

Let r

be a vector of communications variables associ- ated with link l. In general, the capacity c

depends not only on r

, but also on communications variables allo- cated to other links (due to interferences). However, in this paper we will focus on the case where the link ca- pacity is only a function of local resource allocation r

, i.e., c

= φ

(θ

). For example, communication systems with time-division and frequency-division multiple ac- cess (FDMA) fit into this model. We use the following generic model to relate the vector of total traffic t and the vector of communications variables r:

t

≤ c

= φ

(r

), l = 1, . . . , L

F r ¹ g, r º 0 (4)

We make the following assumptions about this model:

• The functions φ

are concave and monotone in- creasing in r

. This implies that the first set of constraints are jointly convex in t and r.

• The second set of constraints describe resource limits, such as the total available transmitting power for the links outgoing from the same node.

Capacity formulas of many important communication channel models satisfy the convexity and monotonic- ity assumptions of the generic model (see, e.g., [6, 7]).

Here we will only illustrate how the Gaussian broad- cast channel with FDMA fits into this framework.

3.1 Gaussian broadcast channel with FDMA In this channel model, the transmitters at node n send data to receivers at the end nodes of its outgoing links.

The outgoing links l ∈ O(n) are assigned disjoint fre- quency bands with bandwidths W

≥ 0 and powers P

≥ 0. The receivers at the end of the links are subject to independent additive white Gaussian noises with power spectral densities σ

. The classical Shan- non capacity formula (see, e.g., [6]) relates the capacity c

and the communications variables r

= (P

, W

) by

c

= φ

(P

, W

) = W

log

µ

1 + P

σ

W

¶

(5)

It can be easily verified that φ

is concave and mono-

tone increasing in the variables (P

, W

). So (5) is in

the generic form of the first set of constraints in (4).

The communications resource limits are X

l∈O(i)

P

≤ P

, X

l∈O(n)

W

≤ W

,

which have the generic form of total resource limits (the second set of constraints) in (4).

3.2 The resource allocation problem

In wireless communication systems, many resource al- location problems can be written in the form of max- imizing a weighted sum of communication rates (as- sume that they are concave functions of resources), i.e.,

maximize P

l

w

c

= P

l

w

φ

(r

)

subject to F r ¹ g, r º 0 (6) where w

are nonnegative weights. For example, with the Gaussian broadcast channel in section 3.1, we can allocate both the power and bandwidth to maximize the total communication rate.

Many specialized algorithms have been developed for problem (6) by exploiting its structure. For example, if there is only one total resource limit, then it can be solved by the classical water-filling algorithm (e.g., [6]).

Actually, water-filling is the one-dimensional version of the dual decomposition method in section 6.

4 The SRRA problem

Combining the network flow model and communica- tions model described in the previous two sections, we now formulate the SRRA problem. We will first present the maximum-utility version of the problem and then discuss many variations of it.

4.1 Maximum-utility SRRA

Consider the operation of a wireless network described by the network flow model (3) and communications model (4). We assume that the utility of each source rate s

(n 6= d) is a concave and increasing function U

(·). Then the maximum-utility SRRA problem is

maximize X

d

X

n,n6=d

U

(s

)

subject to Ax

= s

, d = 1, . . . , D x

º 0, d = 1, . . . , D t

= P

d

x

, l = 1, . . . , L t

≤ φ

(r

), l = 1, . . . , L F r ¹ g, r º 0.

(7)

Here the optimization variables are the network flow variables x, s, t and the communications variables r.

This is a convex optimization problem and it can be solved efficiently by, for example, general interior point methods (see, e.g., [8, 9]). Moreover, in the above model, the matrices A and F are sparse and highly structured, and far more efficient algorithms can be developed by exploiting the problem structure.

4.2 Variations of the SRRA problem

Minimum power SRRA. Given the source vectors s

to be supported by the network, it is desirable to find the joint routing and resource allocation that min- imizes the total transmit power used by the network:

minimize w

r

subject to the constraints in (7)

where w

= 1 if r

is a power variable and w

= 0 otherwise. Many variations, such as minimizing the maximum power used by any node, or minimizing the total bandwidth required to support the desired traffic, can be handled similarly.

Accounting for delays. One of the most common cost functions in communication network literature is the total delay function f

(x, t) = P

l tl

cl−tl

, which is convex if the capacities c

are fixed. The goal is to minimize this function by selecting the routing vari- ables x, when the source vectors s (the throughput of the network) are given. But this function is not jointly convex in t and r when c

is substituted by φ

(r

). An- other cost function with similar qualitative delay prop- erties is the maximum link utilization (see, e.g., [1])

f (t, r) = max

l

t

φ

(r

) .

This function is quasi-convex, and we can use it as the cost function to be minimized in the SRRA problem.

5 Numerical example

Now we consider a randomly generated wireless net- work with 50 nodes (see figure 1). The nodes are uni- formly distributed in the unit square [0, 1]×[0, 1]. Two nodes can communicate to each other if their distance is smaller than 0.25 (the graph is strongly connected).

The network has 340 links (170 double direction links shown in the figure). We randomly choose five source and destination nodes, labeled S

, . . . , S

in figure 1.

S₁

S₂

S₃

S₄ S₅

Figure 1: Topology of a randomly generated network.

In this example, we assume that the bandwidth al- location is fixed and there is no interference among links (using FDMA). We consider the joint routing and power allocation problem. Let P

be the power allo- cated to link l. Each node has a total power constraint for all its outgoing links, i.e., P

l∈O(n)

P

≤ P

= 100

S₁

S₂

S₃

S₄ S₅

(a) Routes to node S1 S₁

S₂

S₃

S₄ S₅

(b) Routes to node S2

S₁

S₂

S₃

S₄ S₅

(c) Routes to node S3 S₁

S₂

S₃

S₄ S₅

(d) Routes to node S4 S₁

S₂

S₃

S₄ S₅

(e) Routes to node S5 S₁

S₂

S₃

S₄ S₅

(f) Power allocation

Figure 2: Optimal routing and power allocation

for n = 1, . . . , N . Let y

be the distance between the two end nodes of link l. We use an inverse-square path- loss model: power at the receiver is given by P

(y

/y

)

, where y

= min

y

is the reference distance. The noise power σ

at each receiver is uniformly distributed on [0.01, 0.1]. We use the following link capacity function and source utility function for problem (7):

φ

(P

) = log

³ 1 +

³ y

y

´

P

σ

´

, U

(s

) = log s

.

We solved the SRRA problem (7) using the dual decomposition method described in section 6. Fig- ure 2(a)–2(e) show the optimal routing tables for the five destinations S

, . . . , S

respectively, and figure 2(f) shows the optimal power allocation. In all these fig- ures, the thickness of each link is roughly proportional to the associated flow variable or the power allocation.

Table 1 shows the source and sink flows which achieve the maximum total utility 17.27. To compare with the SRRA approach, we also solved a maximum utility routing problem with uniform power allocation, where all nodes distribute its total power evenly to its outgo- ing links, and the results are shown in table 2, which

i d = 1 d = 2 d = 3 d = 4 d = 5

1 -3.88 1.11 0.92 1.12 1.13

2 1.03 -16.05 2.93 6.98 6.97

3 0.84 2.69 -9.43 2.69 2.77

4 0.96 4.80 2.46 -18.23 4.80 5 1.05 7.45 3.12 7.44 -15.67 Table 1: Source vectors s

with SRRA i d = 1 d = 2 d = 3 d = 4 d = 5

1 -2.26 1.03 0.88 1.01 1.37

2 0.56 -13.95 1.73 9.59 5.92

3 0.54 2.07 -6.61 1.97 4.14

4 0.54 6.70 1.55 -16.34 4.20 5 0.62 4.15 2.45 3.77 -15.63 Table 2: s

with uniform power allocation

give the maximum total utility 12.77. We see that SRRA gives a 35% improvement of performance.

6 The dual decomposition method

In the SRRA problem (7) (the primal problem), the network flow variables x, s, t and the communications variables r are coupled only through the capacity con- straints t

≤ φ

(r

). We exploit this almost separable structure via dual decomposition (see, e.g., [10, 9]).

6.1 Formulation of the dual problem

We first form the partial Lagrangian, by introducing Lagrange multipliers p ∈ R

only for the L coupling constraints t

≤ φ

(r

). This results in

L(x, s, t, r, p) = X

d

X

n,n6=d

U

(s

)− X

l

p

(t

− φ

(r

))

= X

d

³ X

n,n6=d

U

(s

) − X

l

p

x

´

+ X

l

p

φ

(r

).

We have eliminated the variables t

using t

= P

d

x

in the above expression and will write the Lagrangian as L(x, s, r, p) henceforth. The dual function, i.e., the objective function of the dual problem, is defined as

V (p) = sup

x,s,r

 

 L(x, s, r, p)

¯¯ ¯¯

¯

Ax

= s

, x

º 0 d = 1, . . . , D

F r ¹ g, r º 0

 

 . One immediate observation is that the dual function can be evaluated separately in the network flow vari- ables x, s, t and the communications variables r, i.e.,

V (p) = V

(p) + V

(p),

where V

(p) can be computed by solving the problem maximize X

d

µ X

n,n6=d

U

(s

) − X

l

p

x

¶

subject to Ax

= s

, x

º 0, (8)

d = 1, . . . , D

and V

(p) can be computed by solving the problem maximize P

l

p

φ

(r

)

subject to F ¹ g, r º 0. (9) We call (8) the network flow subproblem and (9) the resource allocation subproblem (same as (6)). Both are convex problems. Actually (8) can be completely decomposed into D single-commodity flow problems.

The dual problem associated with the primal (7) is minimize V (p) = V

(p) + V

(p)

subject to p º 0. (10)

This is a convex optimization problem since V is a con- vex function. Assuming that Slater’s condition for con- straint qualification holds, strong duality holds (see, e.g., [9, 10]), i.e., the optimal values of the dual prob- lem (10) and the primal problem (7) are equal. More- over, we can solve the primal problem via the dual.

6.2 Solve SRRA problem via the dual

Notice that the objective function of the primal prob- lem (7) is not strictly convex. So the dual function V (p) is usually non-differentiable (see, e.g., [10]). Ef- fective methods for solving non-differentiable optimiza- tion problems include the subgradient methods and cutting plane methods, both of which need to compute V (p) and a subgradient of it at a given p º 0.

Compute a subgradient. A subgradient of the con- vex function V at p is a vector h ∈ R

such that

V (q) ≥ V (p) + h

(q − p) (11) for all q. It is a generalization of derivative of differen- tiable functions (see, e.g., [11]). Given a dual variable p º 0, let x

(p), s

(p), t

(p) be one optimal solution to the network flow subproblem (8) and r

(p) be one opti- mal solution to the resource allocation subproblem (9) (they may not be unique). Then a subgradient h of V at p is readily given by (may not be unique either)

h

= φ

(r

(p)) − t

(p), l = 1, . . . , L. (12) One can verify this directly from the definition of V (p).

Recover the primal optimal solution. Now sup- pose that we have numerically solved the dual prob- lem (10), and obtained an optimal dual variable p

(see section 6.3 and 6.4). The corresponding solutions x

(p

), s

(p

), t

(p

) and r

(p

) to the two subprob- lems (8) and (9), however, may not be primal feasi- ble. In particular, they usually don’t not satisfy the capacity constraint t ¹ P

d

x

, which was relaxed when forming the dual problem. This is a typical phe- nomenon for problems with non-strictly convex primal objective functions, where the primal optimal solution is usually a nontrival convex combination of the ex- treme subproblem solutions (see, e.g., [10] chapter 6).

One way to overcome this difficulty is to add a small

strictly-convex regularization term to the primal objec- tive function. For example, we added a small quadratic term of x to the utility function in the example of sec- tion 5. This approach is closely related to augmented Lagrangian and proximal point methods (e.g., [12]).

Next we discuss two algorithms for solving the dual problem (10): the subgradient methods and the ana- lytic center cutting-plane method (ACCPM).

6.3 The subgradient methods

In subgradient methods, we start with an initial point p

. At each iteration k = 1, 2, 3, . . ., we compute the dual function V (p

) and a supergradient h

(see Equation (12)), then update the dual variable by

p

= £

p

− α

h

¤

+

(13)

where [·]

denotes projection onto the nonnegative or- thant, and α

is a positive scalar stepsize. There are many ways to select the stepsizes in subgradient meth- ods. One simple condition for convergence is (e.g.,[11])

k→0

lim α

= 0, and X

α

= ∞.

In the example of section 5, we used the stepsize rule α

= β/k. Figure 3 shows the dual objective function versus the iteration number for β = 0.1 and β = 0.2.

Notice that the subgradient component h

can be ob- tained locally at link l based on its own traffic t

and available capacity φ

(r

). So the subgradient methods can be implemented distributedly at each link, without a central coordinator. Distributed algorithms based on the subgradient methods have been developed for rate control problems in data networks (e.g., [13, 14]).

For extensive accounts of the subgradient methods, as well as many variations, see, e.g., [11, 10].

0 100 200 300 400 500

−100

−50 0 50 100

150 subgradient method β=0.1 subgradient method β=0.2 ACCPM: dual objective ACCPM: lower bound

iteration number

Figure 3: Dual function value versus iteration number

6.4 The analytic center cutting-plane method ACCPM can be viewed as a localization method where we refine the localization polyhedron at each iteration based on a subgradient computed at its analytic center.

Here we will describe ACCPM in the epigraph space

z = (p, v) ∈ R

. Let P = {z | Az ¹ b} be a bounded

polyhedron, where A ∈ R

and b ∈ R

, then its analytic center is defined as

acent( P) = arg max

z∈P

X

log ¡

b

− a

z ¢

where a

is the ith the row of the matrix A.

We start with a polyhedron P

= {z | A

z ¹ b

} which is bounded and contains the optimal solution z

= (p

, V

). For example, it can be a box in R

P

= {z = (p, v) | 0 ¹ p ¹ p, v ≤ v ≤ v}

where v and v are known lower and upper bounds for the optimal value V

, and p is a (component-wise) up- per bound for p

. Then ACCPM can be outlined as

given P

and a required tolerance ² > 0.

k := 0.

repeat

1. Compute the analytic center z

= acent( P

), and obtain a lower bound v from duality.

2. Let p

be the vector of the first L components of z

, and compute V (p

) and a supergradient h

. Then (p

, V

) must lie in the halfspace

H

= {(p, v) | v ≥ V (p

) + h

(p − p

) }.

3. Form the polyhedron P

= P

∩ H

, and update the upper bound v := min {v, V (p

) }.

4. If v − v ≤ ², quit; else, let k := k + 1.

For computational details and convergence analysis of ACCPM, see [15, 9] and references therein.

We applied ACCPM to solve the dual of the SRRA problem in section 5. The dual objective function and the lower bound are also ploted in figure 3. Com- pared to the subgradient methods, ACCPM usually converges faster. However, in ACCPM, all previous computed subgradients are needed to form the local- ization set at each iteration, and we need a central coordinator to compute the analytic center.

7 Conclusions

We considered the problem of simultaneous optimal routing and resource allocation for wireless data net- works. The network routing problem and the resource allocation problem interact through the capacity con- straints of communication links. Using a multicom- modity network flow model and assuming that the link capacities are concave functions of the associated re- source variables, we formulated the SRRA problem as a convex optimization problem. We exploited the sep- arable structure of the SRRA problem via dual decom- position, and solved the dual problem by the subgradi- ent methods and analytic center cutting-plane method.

Acknowledgement

The authors are grateful to Haitham Hindi and Andrea Goldsmith for helpful discussions. This research was sponsored in part by AFOSR grant F49620-01-1-0365, by DARPA contracts F33615-99-C-3014 and MDA972- 02-1-0004, and by MARCO contract 2001-CT-888.

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